Khovanov homology theories and their applications

0.52 MB·English

by  Alexander Shumakovitch

KHOVANOV HOMOLOGY THEORIES
AND THEIR APPLICATIONS
ALEXANDERSHUMAKOVITCH
1
1 To my teacher and advisor, Oleg Yanovich Viro,
0 on the occasion of his 60th birthday
2

n Abstract. This is an expository paper discussing various versions of Kho-
a vanov homology theories, interrelations between them, their properties, and
J theirapplicationstootherareasofknottheoryandlow-dimensionaltopology.

8
2

] 1. Introduction
T
G Khovanovhomologyisaspecialcaseofcategorification,anovelapproachtocon-
. struction of knot (or link) invariants that is being actively developed over the last
h
decade after a seminal paper [Kh1] by Mikhail Khovanov. The idea of categorifica-
t
a tionistoreplaceaknownpolynomialknot(orlink)invariantwithafamilyofchain
m
complexes, such that the coefficients of the original polynomial are the Euler char-
[ acteristics of these complexes. Although the chain complexes themselves depend

1 heavilyonadiagramthatrepresentsthelink,theirhomologydependontheisotopy
v class of the link only. Khovanov homology categorifies the Jones polynomial [J].
4 Morespecifically,letLbeanorientedlinkinR3 representedbyaplanardiagram
1 DandletJ (q)beaversionoftheJonespolynomialofLthatsatisfiesthefollowing
L
6
identities (called the Jones skein relation and normalization):
5
. −q−2J (q)+q2J (q)=(q−1/q)J (q); J (q)=q+1/q. (1.1)
1
0 + − 0
1 The skein relation should be understood as relating the Jones polynomials of three
1 links whose planar diagrams are identical everywhere except in a small disk, where
:
v they are different as depicted in (1.1). The normalization fixes the value of the
i Jones polynomial on the trivial knot. J (q) is a Laurent polynomial in q for every
X L
link L and is completely determined by its skein relation and normalization.
r
a In [Kh1] Mikhail Khovanov assigned to D a family of Abelian groups Hi,j(L)
whose isomorphism classes depend on the isotopy class of L only. These groups
are defined as homology groups of an appropriate (graded) chain complex Ci,j(D)
with integer coefficients. Groups Hi,j(L) are nontrivial for finitely many values
of the pair (i,j) only. The gist of the categorification is that the graded Euler
characteristic of the Khovanov chain complex equals J (q):
L
(cid:88)
J (q)= (−1)iqjhi,j(L), (1.2)
L
i,j
where hi,j(L) = rk(Hi,j(L)), the Betti numbers of H. The reader is referred to
Section 2 for detailed treatment (see also [BN1, Kh1]).
TheauthorispartiallysupportedbyNSFgrantDMS–0707526.
1
2 A.SHUMAKOVITCH
0 1 2 3
9 1
7 1
2
5 1
3 1
1 1
Figure 1. Right trefoil and its Khovanov homology
In our paper we also make use of another version of the Jones polynomial, de-
notedJ(cid:101)L(q),thatsatisfiesthesameskeinrelation(1.1)butisnormalizedtoequal1
on the trivial knot. For the sake of completeness, we also list the skein relation for
the original Jones polynomial, V (t), from [J]:
L
t−1V (t)−tV (t)=(t1/2−t−1/2)V (t); V (t)=1. (1.3)
+ − 0
We note that J (q) ∈ Z[q,q−1] while V (t) ∈ Z[t1/2,t−1/2]. In fact, the terms of
L L
V (t) have half-integer (resp. integer) exponents if L has even (resp. odd) number
L
of components. This is one of the main motivations for our convention (1.1) to be
differentfrom(1.3). WealsowanttoensurethattheJonespolynomialofthetrivial
link has only positive coefficients. The different versions of the Jones polynomial
are related as follows:
JL(q)=(q+1/q)J(cid:101)L(q), J(cid:101)L(−t1/2)=VL(t), VL(q2)=J(cid:101)L(q) (1.4)
Another way to look at the Khovanov’s identity (1.2) is via the Poincar´e poly-
nomial of the Khovanov homology:
(cid:88)
Kh (t,q)= tiqjhi,j(L). (1.5)
L
i,j
With this notation, we get
J (q)=Kh (−1,q). (1.6)
L L
1.A. Example. Consider the right trefoil K. Its non-zero homology groups are
tabulated in Figure 1, where the i-grading is represented horizontally and the j-
grading vertically. The homology is non-trivial for odd j-grading only and, hence,
even rows are not shown in the table. A table entry of 1 or 1 means that the
2
correspondinggroupisZorZ ,respectively(onecanfindamoreinterestingexam-
2
ple in Figure 10). In general, an entry of the form a,b would correspond to the
2
group Za⊕Zb. For the trefoil K, we have that H0,1(K) (cid:39) H0,3(K) (cid:39) H2,5(K) (cid:39)
2
H3,9(K)(cid:39)Z and H3,7(K)(cid:39)Z . Therefore, Kh (t,q)=q+q3+t2q5+t3q9. On
2 K
theotherhand,theJonespolynomialofK equalsV (t)=t+t3−t4. Relation(1.4)
K
implies that J (q)=(q+1/q)(q2+q6−q8)=q+q3+q5−q9 =Kh (−1,q).
K K
Without going into details, we note that the initial categorification of the Jones
polynomialbyKhovanovwasfollowedwithaflurryofactivity. Categorificationsof
KHOVANOV HOMOLOGY THEORIES AND THEIR APPLICATIONS 3
the colored Jones polynomial [Kh3, BW] and skein sl(3) polynomial [Kh4] were
based on the original Khovanov’s construction. Matrix factorization technique
was used to categorify the sl(n) skein polynomials [KhR1], HOMFLY-PT poly-
nomial [KhR2], Kauffman polynomial [KhR3], and, more recently, colored sl(n)
polynomials [Wu, Y]. Ozsv´ath, Szab´o and, independently, Rasmussen used a com-
pletely different method of Floer homology to categorify the Alexander polyno-
mial [OS2, Ra1]. Ideas of categorification were successfully applied to tangles,
virtual links, skein modules, and polynomial invariants of graphs.
One of the most important recent development in the Khovanov homology the-
ory is the introduction in 2007 of its odd version by Ozsv´ath, Rasmussen and
Szab´o[ORS]. TheoddKhovanovhomologyequalstheoriginal(even)onemodulo2
and, in particular, categorifies the same Jones polynomial. On the other hand, the
oddandevenhomologytheoriesoftenhavedrasticallydifferentproperties(seeSec-
tions 2.4 and 3 for details). The odd Khovanov homology appears to be one of the
connecting links between Khovanov and Heegaard-Floer homology theories [OS3].
The importance of the Khovanov homology became apparent after a seminal
result by Jacob Rasmussen [Ra2], who used the Khovanov chain complex to give
thefirstpurelycombinatorialproofoftheMilnorconjecture. Thisconjecturestates
that the 4-dimensional (slice) genus (and, hence, the genus) of a (p,q)-torus knot
equals (p−1)(q−1). ItwasoriginallyprovedbyKronheimerandMrowka[KM1]using
2
the gauge theory in 1993.
There are numerous other applications of Khovanov homology theories. They
can be used to provide combinatorial proofs of the Slice-Bennequin Inequality and
give upper bounds on the Thurston-Bennequin number of Legendrian links, detect
quasi-alternating links and find topologically locally-flatly slice knots that are not
smoothly slice. We refer the reader to Section 4 for details.
The goal of this paper is to give an overview of the current state of research
in Khovanov homology. The exposition is mostly self-contained and no advanced
knowledge of the subject is required from the reader. We intentionally limit the
scope of our paper to the categorifications of the Jones polynomial only, so as to
keepitssizeundercontrol. Thereaderisreferredtootherexpositorypapersonthe
subject [AKh, Kh6, Ra3] to learn more about the interrelations between different
types of categorifications.
We also pay significant attention to experimental aspects of the Khovanov ho-
mology. As is often the case with new theories, the initial discovery is led by
experiments. ItisespeciallytrueforKhovanovhomology,sinceitcanbecomputed
by hands for a very limited family of knots only. At the moment, there are two
programs [BNG, Sh1] that compute Khovanov homology. The first one was writ-
ten by Dror Bar-Natan and his student Jeremy Green in 2005 and implements the
methods from [BN2]. It works significantly faster for knots with sufficiently many
crossings (say, more than 15) than the older program KhoHo by the author. On the
other hand, KhoHo can compute all the versions of the Khovanov homology that
are mentioned in this paper. It is currently the only program that can deal with
the odd Khovanov homology. Most of the experimental results that are referred to
in this paper were obtained with KhoHo .
This paper is organized as follows. In Section 2 we give a quick overview of
constructionsinvolvedinthedefinitionofvariousKhovanovhomologytheories. We
compare these theories with each other and list their basic properties in Section 3.
4 A.SHUMAKOVITCH
Section 4 is devoted to some of the more important applications of the Khovanov
homology to other areas of low-dimensional topology.
This paper was originally presented at the Marcus Wallenberg Symposium on
PerspectivesinAnalysis, Geometry, andTopologyatStockholmUniversityinMay
of 2008. The author would like to thank all the organizers of the Symposium for
a very successful and productive meeting. He extends his special thanks to Ilia
Itenberg, Burglind J¨oricke, and Mikael Passare, the editors of these Proceedings,
fortheirpatiencewiththeauthor. TheauthorisindebtedtoMikhailKhovanovfor
many advises and enlightening discussions during the work on this paper. Finally,
the author would like to express his deepest gratitude to Oleg Yanovich Viro for
introducinghimtothewonderfulworldoftopology20yearsagoandforcontinuing
to be his guide in this world ever since.
2. Definition of the Khovanov homology
In this section we give a brief outline of various Khovanov homology theories
starting with the original Khovanov’s construction. Our setting is slightly more
general than the one in the Introduction as we allow different coefficient rings, not
only Z.
2.1. Algebraic preliminaries. Let R be a commutative ring with unity. In this
paper, we are mainly interested in the cases when R=Z, Q, or Z .
2
2.1.A. Definition. A Z-graded (or simply graded) R-module M is an R-module
(cid:76)
decomposedintoadirectsumM = M ,whereeachM isanR-moduleitself.
j∈Z j j
The summands M are called homogeneous components of M and elements of M
j j
are called the homogeneous elements of degree j.
2.1.B. Definition. Let M = (cid:76) M be a graded free R-module. The graded
j∈Z j
dimension of M is the power series dim (M) = (cid:80) qjdim(M ) in variable q.
q j∈Z j
If k ∈ Z, the shifted module M{k} is defined as having homogeneous components
M{k} =M .
j j−k
2.1.C. Definition. Let M and N be two graded R-modules. A map ϕ:M →N
is said to be graded of degree k if ϕ(M )⊂N for each j ∈Z.
j j+k
2.1.D. It is an easy exercise to check that dim (M{k})=qkdim (M), dim (M ⊕
q q q
N)=dim (M)+dim (N),anddim (M⊗ N)=dim (M)dim (N),whereM and
q q q R q q
N are graded R-modules. Moreover, if ϕ : M → N is a graded map of degree k(cid:48),
then the shifted map ϕ:M →N{k} is graded of degree k(cid:48)+k. We slightly abuse
the notation here by denoting the shifted map in the same way as the map itself.
2.1.E. Definition. Let (C,d) = ··· −→ Ci−1 d−i→−1 Ci −d→i Ci+1 −→ ··· be a
(co)chain complex of graded free R-modules with graded differentials di having
degree 0 for all i ∈ Z. Then the graded Euler characteristic of C is defined as
χ (C)=(cid:80) (−1)idim (Ci).
q i∈Z q
Remark. One can think of a graded (co)chain complex of R-modules as a bigraded
R-module where the homogeneous components are indexed by pairs of numbers
(i,j)∈Z2.
Let A=R[X]/X2 be the algebra of truncated polynomials. As an R-module, A
isfreelygeneratedby1andX. WeputgradingonAbyspecifyingthatdeg(1)=1
KHOVANOV HOMOLOGY THEORIES AND THEIR APPLICATIONS 5
+
positive crossing positive marker
−
negative crossing negative marker
Figure 2. Positiveand Figure 3. Positiveandnegativemarkersand
negative crossings the corresponding resolutions of a diagram.
and deg(X) = −1†. In other words, A (cid:39) R{1}⊕R{−1} and dim (A) = q+q−1.
q
At the same time, A is a (graded) commutative algebra with the unity 1 and
multiplication m:A⊗A→A given by
m(1⊗1)=1, m(1⊗X)=m(X⊗1)=X, m(X⊗X)=0. (2.1)
A can also be equipped with a coalgebra structure with comultiplication ∆ :
A→A⊗A and counit ε:A→R defined as
∆(1)=1⊗X+X⊗1, ∆(X)=X⊗X; (2.2)
ε(1)=0, ε(X)=1. (2.3)
The comultiplication ∆ is coassociative and cocommutative and satisfies
(m⊗id )◦(id ⊗∆)=∆◦m (2.4)
A A
(ε⊗id )◦∆=id (2.5)
A A
Together with the unit map ι : R → A given by ι(1) = 1, this makes A into a
commutative Frobenius algebra over R [Kh5].
It follows directly fromthe definitions that ι, ε, m, and ∆ aregraded maps with
deg(ι)=deg(ε)=1 and deg(m)=deg(∆)=−1. (2.6)
2.2. Khovanov chain complex. Let L be an oriented link and D its planar
diagram. We assign a number ±1, called sign, to every crossing of D according to
the rule depicted in Figure 2. The sum of these signs over all the crossings of D is
called the writhe number of D and is denoted by w(D).
EverycrossingofD canberesolvedintwodifferentwaysaccordingtoachoiceof
amarker,whichcanbeeitherpositiveornegative,atthiscrossing(seeFigure3). A
collectionofmarkerschosenateverycrossingofadiagramDiscalleda(Kauffman)
stateofD. Foradiagramwithncrossings,thereare,obviously,2n differentstates.
Denotebyσ(s)thedifferencebetweenthenumbersofpositiveandnegativemarkers
in a given state s. Define
w(D)−σ(s) 3w(D)−σ(s)
i(s)= , j(s)= . (2.7)
2 2
†Wefollowtheoriginalgradingconventionfrom[Kh1]and[BN1]here. Itisdifferentbyasign
fromtheonein[AKh].
6 A.SHUMAKOVITCH
A⊗A A A⊗A
A
+
m −
+
∆ −
m(1⊗1)=1, m(1⊗X)=m(X⊗1)=X, m(X⊗X)=0
∆(1)=1⊗X+X⊗1, ∆(X)=X⊗X
Figure 4. Diagram resolutions corresponding to adjacent states
and maps between the algebras assigned to the circles
Since both w(D) and σ(s) are congruent to n modulo 2, i(s) and j(s) are always
integer. For a given state s, the result of the resolution of D at each crossing
according to s is a family D of disjointly embedded circles. Denote the number of
s
these circles by |D |.
s
For each state s of D, let A(s) = A⊗|Ds|{j(s)}. One should understand this
construction as assigning a copy of algebra A to each circle from D , taking the
s
tensor product of all of these copies, and shifting the grading of the result by j(s).
Byconstruction,A(s)isagradedfreeR-moduleofgradeddimensiondim (A(s))=
q
qj(s)(q+q−1)|Ds|. LetCi(D)=(cid:76) A(s)foreachi∈Z. InordertomakeC(D)
i(s)=i
into a graded complex, we need to define a (graded) differential di : Ci(D) →
Ci+1(D)ofdegree0. Butevenbeforethisdifferentialisdefined, the(graded)Euler
characteristic of C(D) makes sense.
2.2.A. Lemma. The graded Euler characteristic of C(D) equals the Jones polyno-
mial of the link L. That is, χ (C(D))=J (q).
q L
(cid:88)
Proof. χ (C(D))= (−1)idim (Ci(D))
q q
i∈Z
(cid:88) (cid:88)
= (−1)i dim (A(s))
q
i∈Z i(s)=i
(cid:88)
= (−1)i(s)qj(s)(q+q−1)|Ds|
s
=(cid:88)(−1)w(D)2−σ(s)q3w(D)2−σ(s)(q+q−1)|Ds|.
s
LetusforgetforamomentthatAdenotesanalgebraand(temporarily)usethis
letter for a variable. Substituting (−A−2) instead of q and noticing that w(D) ≡
σ(s) (mod 2), we arrive at
(cid:88)
χ (C(D))=(−A)−3w(D) Aσ(s)(−A2−A−2)|Ds| =(−A2−A−2)(cid:104)L(cid:105) ,
q N
s
where (cid:104)L(cid:105) is the normalized Kauffman bracket polynomial of L (see [K] for de-
N
tails). The normalized bracket polynomial of a link is related to the bracket poly-
nomial of its diagram as (cid:104)L(cid:105) = (−A)−3w(D)(cid:104)D(cid:105). Kauffman proved in [K] that
N
(cid:104)L(cid:105) equals the Jones polynomial V (t) of L after substituting t−1/4 instead of A.
N L
The relation (1.4) between V (t) and J (q) completes our proof. (cid:3)
L L
KHOVANOV HOMOLOGY THEORIES AND THEIR APPLICATIONS 7
Let s and s be two states of D that differ at a single crossing, where s has
+ − +
a positive marker while s has a negative one. We call two such states adjacent.
−
In this case, σ(s ) = σ(s )−2 and, consequently, i(s ) = i(s )+1 and j(s ) =
− + − + −
j(s )+1. Consider now the resolutions of D corresponding to s and s . One
+ + −
can readily see that D is obtained from D by either merging two circles into
s− s+
one or splitting one circle into two (see Figure 4). All the circles that do not pass
through the crossing at which s and s differ, remain unchanged. We define
+ −
d : A(s ) → A(s ) as either m⊗id or ∆⊗id depending on whether the
s+:s− + −
circles merge on split. Here, the multiplication or comultiplication is performed on
thecopiesofAthatareassignedtotheaffectedcircles,asonFigure4,whiled
s+:s−
acts as identity on all the A’s corresponding to the unaffected ones. The difference
in grading shift between A(s ) and A(s ) and (2.6) ensure that deg(d ) = 0
+ − s+:s−
by 2.1.D.
We need one more ingredient in order to finish the definition of the differential
on C(D), namely, an ordering of the crossings of D. For an adjacent pair of states
(s ,s ), define ξ(s ,s ) to be the number of the negative markers in s (or s )
+ − + − + −
that appear in the ordering of the crossings after the crossing at which s and s
+ −
differ. Finally, let di = (cid:80)(s+,s−)(−1)ξ(s+,s−)ds+:s−, where (s+,s−) runs over all
adjacent pairs of states with i(s ) = i. It is straightforward to verify [Kh1] that
+
di+1◦di =0 and, hence, d:C(D)→C(D) is indeed a differential.
2.2.B. Definition (Khovanov, [Kh1]). The resulting (co)chain complex C(D) =
··· −→ Ci−1(D) d−i→−1 Ci(D) −d→i Ci+1(D) −→ ··· is called the Khovanov chain
complex of the diagram D. The homology of C(D) with respect to d is called the
Khovanov homology of L and is denoted by H(L). We write C(D;R) and H(L;R)
if we want to emphasize the ring of coefficients that we work with. If R is omitted
from the notation, integer coefficients are assumed.
2.2.C. Theorem (Khovanov, [Kh1], see also [BN1]). The isomorphism class of
H(L;R) depends on the isotopy class of L only and, hence, is a link invariant.
In particular, it does not depend on the ordering chosen for the crossings of D.
H(L;R) categorifies J (q), a version of the Jones polynomial defined by (1.1).
L
Remark. OnecanthinkofC(D;R)asabigraded(co)chaincomplexCi,j(D;R)with
a differential of bidegree (1,0). In this case, i is the homological grading of this
complex, and j is its q-grading, also called the Jones grading. Correspondingly,
H(L;R) can be considered to be a bigraded R-module as well.
2.2.D. Let #L be the number of components of a link L. One can check that
j(s)+|D | is congruent modulo 2 to #L for every state s. It follows that C(D;R)
s
has non-trivial homogeneous components only in the degrees that have the same
parityas#L. Consequently, H(L;R)isnon-trivialonlyintheq-gradingswiththis
parity (see Example 1.A).
2.2.E. Example. Figure 5 shows the Khovanov chain complex for the Hopf link
withtheindicatedorientation. Thediagramhastwopositivecrossings,soitswrithe
number is 2. Let s be the four possible resolutions of this diagram, where each
±±
“+” or “−” describes the sign of the marker at the corresponding crossings. The
chosenorderingofcrossingsisdepictedbynumbersplacednexttothem. Bylooking
at Figure 5, one easily computes that A(s ) = A⊗2{2}, A(s ) = A(s ) =
++ +− −+
8 A.SHUMAKOVITCH
σ(s−+)=0 A{3}
i(s )=1 − σ(s )=−2
−+ −−
j(s )=3 i(s )=2
−+ −−
j(s )=4
m −−
+
∆
A⊗A{2} (A⊗A){4}
 1  +
−
(cid:77)
C =
 
2 −
+ ∆
−
σ(s++)=2 + m
w(D)=2
i(s )=0
++
j(s )=2
++
σ(s )=0 −
+−
A{3}
i(s )=1
+−
j(s )=3
+−
(cid:16) d0 d1 (cid:17)
C(D) = C0(D) C1(D) C2(D)
Figure 5. Khovanov chain complex for the Hopf link
A{3}, and A(s ) = A⊗2{4}. Correspondingly, C0(D) = A(s ) = A⊗2{2},
−− ++
C1(D) = A(s )⊕A(s ) = (A⊕A){3}, and C2(D) = A(s ) = A⊗2{4}. It is
+− −+ −−
convenient to arrange the four resolutions in the corners of a square placed in the
planeinsuchawaythatitsdiagonalfroms tos ishorizontal. Thentheedges
++ −−
of this square correspond to the maps between the adjacent states (see Figure 5).
We notice that only one of these maps, namely the one corresponding to the edge
from s to s , comes with the negative sign.
+− −−
In general, 2n resolutions of a diagram D with n crossings can be arranged into
an n-dimensional cube of resolutions, where vertices correspond to the 2n states
of D. The edges of this cube connect adjacent pairs of states and can be oriented
from s+ to s−. Every edge is assigned either m or ∆ with the sign (−1)ξ(s+,s−),
as described above. It is easy to check that this makes each square (that is, a
2-dimensional face) of the cube anti-commutative (all squares are commutative
without the signs). Finally, the differential di restricted to each summand A(s)
with i(s) = i equals the sum of all the maps assigned to the edges that originate
at s.
2.3. Reduced Khovanov homology. Let, as before, D be a diagram of an ori-
ented link L. Fix a base point on D that is different from all the crossings. For
each state s, we define A(cid:101)(s) in almost the same way as A(s), except that we assign
XA instead of A to the circle from the resolution D of D that contains that base
s
point. That is, A(cid:101)(s)=(cid:0)(XA)⊗A⊗(|Ds|−1)(cid:1){j(s)}. We can now build the reduced
Khovanov chain complex C(cid:101)(D;R) in exactly the same way as C(D;R) by replacing
A with A(cid:101) everywhere. The grading shifts and differentials remain the same. It
is easy to see that C(cid:101)(D;R) is a subcomplex of C(D;R) of index 2. In fact, it is
KHOVANOV HOMOLOGY THEORIES AND THEIR APPLICATIONS 9
the image of the chain map C(D;R) → C(D;R) that acts by multiplying elements
assigned to the circle containing the base point by X.
2.3.A. Definition(Khovanov[Kh2],cf.2.2.B). ThehomologyofC(cid:101)(D;R)iscalled
the reduced Khovanov Homology of L and is denoted by H(cid:101)(L;R). It is clear from
the construction of C(cid:101)(D;R) that its graded Euler characteristic equals J(cid:101)L(q).
2.3.B. Theorem (Khovanov[Kh2],cf.2.2.C). The isomorphism class of H(cid:101)(L;R)
is a link invariant that categorifies J(cid:101)L(q), a version of the Jones polynomial defined
by (1.1) and (1.4). Moreover, if two base points are chosen on the same component
of L, then the corresponding reduced Khovanov homologies are isomorphic. On the
other hand, H(cid:101)(L;R) might depend on the component of L that the base point is
chosen on.
Although C(cid:101)(D;R) can be determined from C(D;R), it is in general not clear
how H(L;R)and H(cid:101)(L;R) arerelated. Thereare several examplesof pairsof knots
(the first one being 14n and 15n ‡) that have the same rational Khovanov
9933 129763
homology, but different rational reduced Khovanov homology. No such examples
are known for homologies over Z among all prime knots with at most 15 crossings.
On the other hand, it is proved that H(L;Z2) and H(cid:101)(L;Z2) determine each other
completely.
2.3.C. Theorem ([Sh2]). H(L;Z2) (cid:39) H(cid:101)(L;Z2)⊗Z2 AZ2. In particular, H(cid:101)(L;Z2)
does not depend on the component that the base point is chosen on.
Remark. XA (cid:39) R{0} as a graded R-module. It follows that C(cid:101) and H(cid:101) are non-
trivial only in the q-gradings with parity different from that of #L, the number of
components of L (cf. 2.2.D).
2.4. Odd Khovanov homology. In 2007, Ozsv´ath, Rasmussen and Szab´o in-
troduced [ORS] an odd version of the Khovanov homology. In their theory, the
nilpotent variables X assigned to each circle in the resolutions of the link diagram
(seeSection2.2)anti-commuteratherthancommute. TheoddKhovanovhomology
equals the original (even) one modulo 2 and, in particular, categorifies the same
Jonespolynomial. Infact,thecorrespondingchaincomplexesareisomorphicasfree
bigradedR-modulesandtheirdifferentialsareonlydifferentbysigns. Ontheother
hand, the resulting homology theories often have drastically different properties.
We define the odd Khovanov homology below.
Let L be an oriented link and D its planar diagram. To each resolution s of
D we assign a free graded R-module Λ(s) as follows. Label all circles from the
resolution D by some independent variables, say, Xs,Xs, ... ,Xs and let V =
s 1 2 |Ds| s
V(Xs,Xs, ... ,Xs ) be a free R-module generated by them. We define Λ(s) =
1 2 |Ds|
Λ∗(Vs), the exterior algebra of Vs. Then Λ(s) = Λ0(Vs)⊕Λ1(Vs)⊕···⊕Λ|Ds|(Vs)
andwegradeΛ(s)byspecifyingΛ(s) =Λk(V )foreach0≤k ≤|D |, where
|Ds|−2k s s
Λ(s) is the homogeneous component of Λ(s) of degree |D |−2k. It is an
|Ds|−2k s
easy exercise for the reader to check that dimq(Λ(s))=dimq(A⊗|Ds|).
JustasinthecaseoftheevenKhovanovhomology,theseR-modulesΛ(s)canbe
arrangedintoann-dimensionalcubeofresolutions. LetCi (D)=(cid:76) Λ(s){j(s}.
odd i(s)=i
‡Here, 14n denotes the non-alternating knot number 9933 with 14 crossings from the
9933
Knotscape knot table [HTh] and 15n is the mirror image of the knot 15n . See also
129763 129763
remarkonpage13.
10 A.SHUMAKOVITCH
+
rotate the arrow by 90◦ clockwise −
Figure 6. Choice of arrows at the diagram crossings
X+ X+ X− X+ X−
1 + 2 1 1 2
m −
odd +
∆ −
odd
Λ∗V(X−)(cid:39)Λ∗V(X+,X+)/(X+−X+)
1 1 2 1 2
modd(α)=α/(X1+−X2+), ∆odd(α)=(X1−−X2−)∧α X−
1
Figure 7. Adjacent states and differentials in the odd Khovanov
chain complex
Then, similarly to Lemma 2.2.A, we have that χ (C (D)) = J (q). In fact,
q odd L
C (D)(cid:39)C(D)asbigradedR-modules. InordertodefinethedifferentialonC ,
odd odd
we need to introduce an additional structure, a choice of an arrow at each crossing
of D that is parallel to the negative marker at that crossing (see Figure 6). There
areobviously2n suchchoices. ForeverystatesonD,weplacearrowsthatconnect
twobranchesofD neareach(former)crossingaccordingtotherulefromFigure6.
s
We now assign (graded) maps m and ∆ to each edge of the cube of res-
odd odd
olutions that connects adjacent states s and s . If s is obtained from s by
+ − − +
merging two circles together, then Λ(s )(cid:39)Λ(s )/(X+−X+), where X+ and X+
− + 1 2 1 2
are the generators of V corresponding to the two merging circles, as depicted in
s+
Figure 7. We define m :Λ(s )→Λ(s ) to be this isomorphism composed with
odd + −
the projection Λ(s )→Λ(s )/(X+−X+).
+ + 1 2
The case when one circle splits into two is more interesting. Let X− and X− be
1 2
the generators of V corresponding to these two circles such that the arrow points
s−
from X− to X− (see Figure 7). Now for each generator X+ of V , we define
1 2 k s+
∆ (X+) = (X− −X−)∧X− where η is the correspondence between circles
odd k 1 2 η(k)
in D and D . While η(1) can equal either 1 or 2, this choice does not affect
s+ s−
∆ (X+) since (X−−X−)∧X− =X−∧X− =−X−∧X− =(X−−X−)∧X−.
odd 1 1 2 2 1 2 2 1 1 2 1
Thisdefinitionmakeseachsquareinthecubeofresolutionseithercommutative,
or anti-commutative, or both. The latter case means that both double-composites
correspondingtothesquarearetrivial. Thisisamajordeparturefromthesituation
that we had in the even case, where each square was commutative. In particular,
it makes the choice of signs on the edges of the cube much more involved.
2.4.A. Theorem (Ozsv´ath–Rasmussen–Szab´o [ORS]). It is possible to assign a
sign to each edge in this (odd) cube of resolutions in such a way that every square